Inverse problem for a multidimensional heat equation with an unknown source function (Q1605613)

The author considers an inverse problem for the equation \(u_t=\Delta u+g_0(x,t)+f_1(x)g_1(x)+f_2(x)g_2(x), x\in D\subset{\mathbb R}^n\), \(0<t<T,\) with unknown functions \(f_1\) and \(f_2\), the initial condition \(u(x,0)=\varphi(x)\), \(x\in \overline{D},\) the boundary condition \(u(x,t)|_{S_T}=\mu(x,t)\), \((x,t)\in S_T=\partial D\times[0,T],\) and the overdetermination conditions of the form \(u(x_0,t)=\kappa(t)\), \(x_0\in D\), \(t\in[0,T]\), \(\int_0^Tu(x,t) dt=\psi(x)\), \(x\in\overline{D}\). Existence and uniqueness conditions for the inverse problem are given.